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When the first column only has a zero we replace is with an $\epsilon$ so we don't divide by zero ahead but divide by the $\epsilon$ then take the limit as it goes to zero.

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meanwhile if an entire row has zeros we use the coefficients of the derivative of the auxiliary polynomial

enter image description here What I can't seem to understand here is why can't we just use the $\epsilon$ method. We won't be dividing by any number except for those in the first column and thus it feels intuitive to just replace the zero in the first column of the entire row of zeros with an epsilon and proceed as normal. Why wouldn't that work?

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